Was any war started by a country (both of its people and leadership) knowing defeat was inevitable?

My memory might serve me wrong, but I seem to recollect quite a few pundits pointing out that nobody in recent memory had ever won a war in Afghanistan when G. W. Bush wanted to go there.

This impression certainly wasn't universal then - and indeed there were quite a few hawks. But the notion that Afghanistan was an unwinnable war was widespread insofar as I can recollect. And sure enough, here we are more than a dozen years later and it still isn't a won war, as feared by many in and out of the US when the US went there.

To answer about Japan's situation in 1941, you're on false premises.

No, people who decide in Japan -the Emperor, the Prime Minister, most of the Navy and Army- did not consider defeat as sure in the war they were about to start. They considered that it was possible to:

• Defeat the European forces in Asia-Pacific because they were already involved in the war in Europe, and two of them (France and the Netherlands) were occupied -> And this was correct
• Defeat the Asiatic Fleet and prevent a fast response from the Americans, this turned into Pearl Harbour and the Philippines attack -> And this indeed worked
• Capture most of the territories needed to supply Japanese's industry-> And this was correct

But they faced some difficulties they did not anticipated:

• Weaknesses of their military forces
• Poor logistics that did not permit to exploit in a timely manner their conquests

And they took poor decisions during the war that ruined their chances to resist:

• Battle of Midway
• Attacks of India during U-Go

The events leading to the Paraguayan War can be interpreted as Paraguay starting a war they could not win.

I'm no expert on this war, but my understanding is that at the time the strongest countries in the region were Brazil and Argentina. Paraguay concluded an alliance with Uruguay, which was another country that was regularly under threat from the two regional hegemons. Intrigue in Uruguay eventually led to Brazilian intervention. One side of the warring Uruguayan factions asked Paraguay for assistance. Paraguay voiced a diplomatic protest to Brazil, which was ignored, so Paraguay attempted to dispatch troops to Uruguay. To do this they needed to cross Argentine territory, and Argentina denied permission. So Paraguay declared war on Argentina, then Brazil. Meanwhile in Uruguay the Brazil-allied faction had won the civil war, and all three powers combined to declare war on Paraguay. The result was a war between a country with 450k people against a combined 11 million people. Paraguay was smashed and a lot of their citizens died.

Although it's obvious that Paraguay cannot win the war, I can't find any sources that stated that Francisco Solano López (Paraguayan president at the time) went to war knowing that he cannot win - but equally, I can't find any sources that state he aimed to win, as well.

I'd say that the Third Punic war might fulfill the requirement. Even though the beginning of the war might be hard to define, Roman demand to people of Carthage to leave the city and go inland to live, in order to burn the city, almost forced Carthage to accept war instead of the relocation. Even though they did not have weapons neither military training. So defeat was almost for sure.

Another potential candidate is the Warsaw ghetto uprising during WWII. In this case the difference is that the getto was not a country, but a nation. But the other requirements of the OP are filled, because people in the ghetto knew they could not win against germans. But it was the inevitable defeat or the extermination camp.

Shortest answer:

No.

• Uncertainty is so important in war that one cannot be sure of defeat either.
• Even if certainty could be imagined here (it cannot!), it is still hard to imagine how it may apply to a whole country, "both its people and its leadership". Only in a war or/and totalitarian propaganda discourse can a such statement be imagined.
• Propaganda can only be optimistic. If a such totalitarian unanimity could be achieved, or simply in order for it to be articulated in propaganda, the goal can only be victory, and not defeat.

Elaborating a bit more:

Defeat and victory are relative terms, depending on the goals, which in turn can be multiple and of different degree of probability. Certainty and uncertainty are decisive factors here. One could start a war that as such has no chance of winning but that might trigger further events (further wars, alliances) that they want to achieve.

Also the issue of "who started first" is a relative one too. Weaker countries may start wars because their position is untenable anyway (they suffer too much or are about to be conquered anyway).

Not all wars are total wars, and that of Japan against the US might have succeeded at least as far as reaching some intermediary goals. Unconditional surrender (as one comment says) was not a necessary scenario initially. Japan might have imagined some kind of a settled peace, just as the Nazis hoped for until very late. Japan wasn't sure it will lose the war, their goal was to destroy the US fleet, and that wasn't an impossible task in fact. They wanted their share of colonial empire (the Nazis too in fact, according to Timothy Snyder), and they could have hoped to have that confirmed in the end anyway on China's expense, like the Nazi's initially hoped to get one in Ukraine (if it wasn't for the stubbornness of the Brits and the Soviet peoples themselves). Japan and Germany wanted to be respected, that is be equal to the other colonial powers, and the war between such powers needn't be motivated by certainties. In the aristocratic imaginary of both imperialist Japanese and Nazi Germans war was the state of normality (like for a medieval knight or a samurai), and that didn't require certitude of victory. It entailed in a way the contrary, the certitude of (hopefully glorious) death: but not of defeat.

Arguably Japan is not a good example here, but from a general perspective one may say that many rebellions or wars of liberation against a much stronger power would qualify. Just like one may lose battles and win the war, or lose the war but win the peace, many intermediary cases are possible.

"Knowing" is also a relative term. What can it mean to "know" you will win? It can only mean "hope", thus fighting is never done "knowing" you'll lose no more than it is done knowing you'll win. - And what can it mean that the leadership knows something but not the people, or that both, or none, knows it? (How can we separate between what the Japanese generals knew and what the "people" did? Was "the people" even supposed to know they'll win or lose? And was a such popular knowledge supposed to count in the decision of starting the war? Isn't that just propaganda?) - Ignorance, like hope, is a big part of "knowing" such things.

From the perspective of yet unborn nation states or of otherwise oppressed peoples it is a big victory even to be able to mount an armed action against the masters (like Serbs, Greeks, Bulgarians against the Ottoman, the native Americans against the US, Algerians against the French, Indians against the British, Caucasus peoples against the Russian, other colonized peoples against the colonizers, Poles against the Russians or the Germans).

Starting a war may equate to proclaiming a new state (status, liberty), and for oppressed peoples that is as important as victory.

In the history of my native Wallachian and Moldavian lands, wars against the Turks had the goal sometimes to trigger an intervention from Hungary or Poland, or just press for the change in the conditions of the dependent status, but that doesn't mean they couldn't have at the same time the maximalist goal of removing that dependence. There are also cases were one could argue that these principalities periodically rebelling against the Turks contributed to their "victory" of not being fully occupied and keeping internal autonomy.

War and peace are also a matter of imagination and ethics. Sometimes the Mongols used to send emissaries before attacking, letting the recipient know that refusing Mongol demands equated to a declaration of war. Thus, it was the Hungarian or Polish kings that were made to appear as rejecting peace. But for these kings surrendering to the Mongols without a fight was ethically but also practically unimaginable. That was because they didn't knew who the Mongols were, but also because they didn't knew how to surrender without a fight! (Mongol brutality and cruelty was a political means of propaganda: of making people understand how one can and must surrender without a fight - the equivalent of the present atomic menace). Therefore they chose to fight the Mongol although the chances of beating them were slim (and although historians have concluded that their slim victories counted for little in the outcome of Mongols not occupying all Europe.)